Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
arXiv:1110.5435v2 [math.DS] 10 Jan 2012 DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION JIAN LI Abstract. In this paper, we set up a general correspondence between the algebra properties of βN and the sets defined by dynamical properties. In particular, we obtain a dynamical characterization of C-sets, where C-sets are the sets satisfying the strong Central Sets Theorem. As an application, we show that Rado systems are solvable in C-sets. 1. Introduction Throughout this paper, let Z, Z+ , N and Q denote the sets of the integers, the non-negative integers, the positive integers and the rational numbers, respectively. Let us recall two celebrated theorems in combinatorial number theory. S Theorem 1.1 (van de Waerden). Let r ∈ N and N = ri=1 Ci . Then there exists some i ∈ {1, 2, . . . , r} such that Ci contains arbitrarily long arithmetic progressions. ∞ For a sequence {xn }∞ n=1 in N, define the finite sums of {xn }n=1 as nX o ) = F S({xn }∞ x : α is a nonempty finite subset of N . n n=1 n∈α A subset F of N is called an IP set if there exists a sequence {xn }∞ n=1 in N such that F S({xn }∞ n=1 ) ⊂ F . S Theorem 1.2 (Hindman). Let r ∈ N and N = ri=1 Ci . Then there exists some i ∈ {1, 2, . . . , r} such that Ci is an IP set. The original proofs of the above two theorems are somewhat complicated by combinatorial methods. In [14, 12] Furstenberg and Weiss found a new way to prove those theorems by topological dynamics methods. A subset F of N is called central if there exists a dynamical system (X, T ), a point x ∈ X, a minimal point y which is proximal to x, and 2010 Mathematics Subject Classification. Primary 37B20; Secondary 37B05, 05D10. Key words and phrases. Central sets, C-sets, Rado system. 1 2 J. LI an open neighborhood U of y such that F = {n ∈ N : T n x ∈ U}. Then van de Waerden Theorem and Hindman Theorem follow from the following result. Theorem 1.3 ([14, 12]). (1) Every central set is an IP set and contains arbitrarily long arithmetic progressions. S (2) Let r ∈ N and N = ri=1 Ci . Then there exists some i ∈ {1, 2, . . . , r} such that Ci is a central set. Before going on, let us recall some notions. We call (S, ·) a compact Hausdorff right topological semigroup if S is endowed with a topology with respect to which S is a compact Hausdorff space and for each t ∈ S the right translation s 7→ s · t is continuous. An idempotent t ∈ S is an element satisfying t · t = t. Ellis-Namakura Theorem says that any compact Hausdorff right topological semigroup contains some idempotent. A subset I of S is called a left ideal of S if SI ⊂ I, a right ideal if IS ⊂ I, and a two sided ideal (or simply an ideal ) if it is both a left and right ideal. A minimal left ideal is the left ideal that does not contain any proper left ideal. Similarly, we can define minimal right ideal and minimal ideal. An idempotent in S is called a minimal idempotent if it is contained in some minimal left ideal of S. Endowing N with the discrete topology, we take the points of the Stone-Čech compactification βN of N to be the ultrafilter on N. Since (N, +) is a semigroup, we extend the operation + to βN such that (βN, +) is a compact Hausdorff right topological semigroup. See [19] for an exhaustive treatment of the algebraic structure on βN. Ellis showed that we can regard (βN, N) as a universal point transitive system ([10]). One may think that there is a nature connection between the algebra properties of βN and the sets defined by dynamical properties. For example, in [5] Bergelson and Hindman showed that Theorem 1.4 ([5]). A subset F of N is central if and only if there exists a minimal idempotent p ∈ βN such that F ∈ p. A subset F of N is called quasi-central if there exists an idempotent p ∈ βN with each element being piecewise syndetic such that F ∈ p. Of course, every quasi-central set is central, but there exists some quasicentral set which is not central ([18]). The authors in [8] showed a dynamical characterization of quasi-central sets. Theorem 1.5 ([8]). A subset F of N is quasi-central if and only if there exists a dynamical system (X, T ), a pair of points x, y ∈ X where for every open neighborhood V of y the set {n ∈ N : T n x ∈ V, T n y ∈ V } is piecewise syndetic, and an open neighborhood U of y such that F = {n ∈ N : T n x ∈ U}. DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 3 A subset F of N is called a D-set if there exists an idempotent p ∈ βN with each element having positive upper Banach density such that F ∈ p. It should be noticed that every quasi-central set is a D-set and there exists some D-set which is not quasi-central ([4]). There is also a dynamical characterization of D-sets. Theorem 1.6 ([4]). A subset F of N is a D-set if and only if there exists a dynamical system (X, T ), a pair of points x, y ∈ X where for every open neighborhood V of y the set {n ∈ N : T n y ∈ V } has positive upper Banach density and (y, y) belongs the orbit closure of (x, y) in the product system (X × X, T × T ), and an open neighborhood U of y such that F = {n ∈ N : T n x ∈ U}, Central sets have substantial combinatorial contents. In order to describe the properties, we first introduce some notations. By Pf (N) we denote the set of all nonempty finite subsets of N. For α, β ∈ Pf (N), we write α < β if max α < minP β. Given a sequence s1 , s2 , . . . in Z or Zm and α ∈ Pf (N) we let sα = n∈α sn and call the family (sα )α∈Pf (N) an IP-system. A homomorphism φ : Pf (N) → Pf (N) is a map such that (1) if α ∩ β = ∅, then φ(α) ∩ φ(β) = ∅ and (2) φ(α ∪ β) = φ(α) ∪ φ(β). Evidently such a homomorphism is determined by φ({i}) S on each i ∈ N, and then φ(α) = i∈α φ({i}). Given an IP-system {sα }, an IP-subsystem is defined by a homomorphism φ : Pf (N) → Pf (N) and forming {sφ(α) } ⊂ {sα }. If r ∈ Z, we shall denote by r̄ (m) the vector (r, . . . , r) ∈ Zm . Proposition 1.7 (Central Sets Theorem [12]). Let F be a central set in N, and for any m ≥ 1, let {sα } be any IP-system in Zm . We can find an IP-subsystem {sφ(α) } and an IP-system {rα } in N such that the (m) vector r̄α + sφ(α) ∈ F m for each α ∈ Pf (N). Recently, the authors in [9, 20] proved a stronger version of the Central Sets Theorem, they call C-sets are the sets satisfying the conclusion of the strong Central Sets Theorem. Here we will not discuss the strong Central Sets Theorem, so we adopt an alternative definition of C-sets. A subset F of N is called a J-set if for every m ∈ N and every IP-system {sα } in Zm there exists r ∈ N and α ∈ Pf (N) such that r̄ (m) + sα ∈ F m . Denote by J the collection of all J-sets. A subset F of N is called a C-set if there exists an idempotent p ∈ βN with each element being J-set such that F ∈ p. Since every positive upper Banach density set is a J-set ([13]), every D-set is a C-set. But there exists a C-set with zero upper Banach density ([17]), so this set is not a D-set. In this paper, we obtain a dynamical characterization of C-sets. 4 J. LI Theorem 1.8. A subset F of N is a C-set if and only if there exists a dynamical system (X, T ), a pair of points x, y ∈ X where for any open neighborhood V of y the set {n ∈ N : T n y ∈ V } is a J-set and (y, y) belongs to the orbit closure of (x, y) in the product system (X × X, T × T ), and an open neighborhood U of y such that F = {n ∈ N : T n x ∈ U}, In [12] Furstenberg used the Central Sets Theorem to show that any central subset of N contains solutions to all Rado systems. Let A = (aij ) be a p × q matrix over Q, the homogeneous system of linear equations A(x1 , . . . , xq )T = 0 S is called partition regular if for every r ∈ N and N = ri=1 Ci , there exists some i ∈ {1, 2, . . . , r} such that the system has a solution (x1 , . . . , xq ) all of whose components lie in Ci . Each such homogeneous system of linear equations is called a Rado system. In [26] Rado characterized when a homogeneous system of linear equations is partition regular. Theorem 1.9 (Rado’s Theorem). Let A = (aij ) be a p × q matrix over Q. Then the system A(x1 , . . . , xq )T = 0 is partition regular if and only if the index set {1, 2, . . . , q} can be divided into l disjoint subsets I1 , I2 , . . . , Il and rational numbers crj may be found for r ∈ {1, . . . , l} and j ∈ I1 ∪ · · · ∪ Ir such that the following relationships are satisfied: X aij = 0, j∈I1 X aij = j∈Il c1j aij , j∈I1 j∈I2 ··· X X aij = X cl−1 j aij . j∈I1 ∪I2 ∪···∪Il−1 Let F be a subset of N, we say Rado systems are solvable in F if every Rado system A(x1 , . . . , xq )T = 0 has a solution (x1 , . . . , xq ) all of whose components lie in F . Furstenberg and Weiss improved Rado’s result by showing that Theorem 1.10. [14, 12] Rado systems are solvable in central sets. Recently, the authors in [3] extended Furstenberg and Weiss’ result to Theorem 1.11. [3] Rado systems are solvable in D-sets. DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 5 In this paper, we use the dynamical characterization of C-sets to show that Theorem 1.12. Rado systems are solvable in C-sets. This paper is organized as follows. In Section 2 we introduce some notations related to Furstenberg families. In Section 3 the basic properties of the Stone-Čech compactification of N are discussed. In Section 4 we set up a general correspondence between the algebra properties of βN and the sets defined by dynamical properties. The dynamical characterizations of quasi-central sets and D-sets are special cases of our results. In Section 5, we investigate the set’s forcing that is the dynamical properties of a point along a subset of N. In Section 6, we consider both addition and multiplication in N and βN. Particularly we show that if F is a quasi-central set or a D-set, then for every n ∈ N both nF and n−1 F are also quasi-central sets or D-sets. In Section 7 using the correspondence which is set up in Section 4 and some properties of J-sets, we obtain a dynamical characterization of C-sets. In Section 8, as an application, we give a topological dynamical proof of the fact that Rado systems are solvable in C-sets. 2. Furstenberg family Let us recall some notations related to a family (for more details see [1]). For the set of positive integers N, denote by P = P(N) the collection of all subsets of N. A subset F of P is called a Furstenberg family (or just family) if it is hereditary upward, i.e., F1 ⊂ F2 and F1 ∈ F imply F2 ∈ F . A family F is called proper if it is a nonempty proper subset of P, i.e., neither empty nor all of P. For a family F , the dual family of F , denote by κF , is {F ∈ P : F ∩ F ′ 6= ∅, ∀F ′ ∈ F }. Sometimes the dual family κF is also denoted by F ∗ . A family F is called a filter when it is a proper family closed under intersection, i.e., if F1 , F2 ∈ F then F1 ∩ F2 ∈ F . A family F is called a filterdual if its dual κF is a filter. It is easy to see that a proper family F is a filterdual if and only if it satisfies the Ramsey Property, i.e., if F1 ∪ F2 ∈ F then either F1 ∈ F or F2 ∈ F . Since κ(κF ) = F , a family F is a filter if and only if κF is a filterdual. Of special interest are filter that are maximal with respect to inclusion. Such a filter is called an ultrafilter. By Zorn’s Lemma every filter is contained in some ultrafilter. For any n ∈ N the family {A ⊂ N : n ∈ A} is an ultrafilter. An ultrafilter formed in this way 6 J. LI is called a principal ultrafilter. Any other ultrafilter is called a nonprincipal ultrafilter. The following two lemmas are basic properties of ultrafilter, see [1, 15, 19] for example. Lemma 2.1. Let F be a filter. Then the following conditions are equivalent: (1) F is an ultrafilter; (2) F = κF ; (3) F is a filterdual; (4) For all F ⊂ N, either F ∈ F or N \ F ∈ F . Lemma 2.2. Let F be a filterdual and A T ⊂ F . If for any finite elements A1 , A2 ,. . ., An in A the intersection ni=1 Ai is in F , then there exists an ultrafilter F ′ such that A ⊂ F ′ ⊂ F . For n ∈ Z and F ⊂ N, denote n + F = {n + m ∈ N : m ∈ F }. A family F is called translation + invariant if n + F ∈ F for every n ∈ Z+ and F ∈ F , translation − invariant if −n + F ∈ F for every n ∈ Z+ and F ∈ F and translation invariant if it is both + and − invariant. Any nonempty collection A of subsets of N naturally generates a family F (A) = {F ⊂ N : F ⊃ A for some A ∈ A}. A collection A of subsets of N is said to have the finite intersection property if the intersection of any finite elements in A is not empty. In this case, the family generated by A is a filter. Let F be a family, the block family of F , denote by bF , is the family consisting of sets F ⊂ N for which there exists some F ′ ∈ F such that for every finite subset W of F ′ one has m + W ⊂ F for some m ∈ Z+ . It is easy to see that F ∈ bF if and only a sequence S∞if there exists ∞ ′ ′ {an }n=1 in Z+ and F ∈ F such that n=1 (an + F ∩ [1, n]) ⊂ F . Clearly, b(bF ) = bF and bF is translation + invariant. Lemma 2.3. [7, 22] If F is a filterdual, then so is bF . Now let us recall some important sets and families. Let Finf be the family of all infinite subsets of Z+ . It is easy to see that its dual family κFinf is the family of all cofinite subsets, denoted by Fcf . A subset F of Z+ is called thick if it contains arbitrarily long runs ∞ of positive S∞ integers, i.e., there exists a sequence {an }n=1 in Z+ such that n=1 (an + [1, n]) ⊂ F ; syndetic if there exists N ∈ N such that [n, n + N] ∩ F 6= ∅ for every n ∈ N; piecewise syndetic if it is the intersection of a thick set and a syndetic set. The families of all thick DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 7 sets, syndetic sets and piecewise syndetic sets are denoted by Ft , Fs and Fps , respectively. It is easy to see that κFs = Ft . Let F be a subset of N, the upper density of F is ¯ ) = lim sup |F ∩ [1, n]| , d(F n n→∞ where | · | denote the cardinality of the set, and the upper Banach density of F is |F ∩ I| BD ∗ (F ) = lim sup |I| |I|→∞ where I is taken over all nonempty finite intervals of N. Using density we can define lots of interesting families. For example, denote Fpud and Fpubd by the family of sets with positive upper density and positive upper Banach density respectively. Denote by Fip and Fcen the family of all IP sets and central sets respectively. We have the following basic property about the familiar families, see [1, 19] for example. Lemma 2.4. (1) Fcen , Fip , Fps , Fpud and Fpubd are filterduals. (2) Fps , Fpud, Fpubd and Fs are translation invariant. (3) bFcf = Ft , bFs = Fps and bFpud = Fpubd. We now introduce the notion of F -limit. Let F be a family and {xn }n∈N be a sequence in a topological space, we say that x is a F limit of {xn } if for every open neighborhood U of x the set {n ∈ N : xn ∈ U} ∈ F . Then Fcf -limit is just the ordinary convergence. It is easy to check that If F is a filter then F -lim xn exists and is unique in every compact Hausdorff space. 3. βN: the Stone-Čech compactification of N Endowing N with the discrete topology, we take the points of the Stone-Čech compactification βN of N to be the ultrafilter on N, the principal ultrafilter being identified with the points in N. For A ⊂ N, let A = {p ∈ βN : A ∈ p}. Then the sets {A : A ⊂ N} forms a basis for the open sets (and a basis for the closed sets) of βN. Since (N, +) is a semigroup, we can extend the operation + to βN as p + q = {F ⊂ N : {n ∈ N : −n + F ∈ q} ∈ p}. Then (βN, +) is a compact Hausdorff right topological semigroup with N contained in the topological center of βN. That is, for each p ∈ βN the map ρp : βN → βN, q 7→ q + p is continuous, and for each n ∈ N the map λn : βN → βN, q 7→ n + q is continuous. It is well known that 8 J. LI S βN hasSa smallest ideal K(βN) = {L : L is a minimal left ideal of βN} = {R : R is a minimal right ideal of βN} ([19, Theorem 2.8]). Lemma 3.1. Let F be a filter. If for every F ∈ F thereTexists some F ′ ∈ F such that −n + F ∈ F for every n ∈ F ′ , then F ∈F F is a closed subsemigroup of βN. T Proof. Since F has finite intersection property, F ∈F F is nonempty. T T Let p, q ∈ F ∈F F , we want to show that p + q ∈ F ∈F F . Let F ∈ F , it suffices to show that F ∈ p + q. For this F , there exists some F ′ ∈ F such that −n + F ∈ F for every n ∈ F ′ . Then F ′ ⊂ {n ∈ N : −n + F ∈ q} and {n ∈ N : −n + F ∈ q} ∈ p. By the definition of “+” in βN we have F ∈ p + q. Lemma 3.2 ([19, Theorem 4.20]). Let A be a collection of subset of N. If A has the finite intersection property and for everyTF ∈ A and n ∈ F there exists F ′ ∈ A such that n + F ′ ⊂ F , then F ∈A F is a closed subsemigroup of βN. For a filterdual F , the hull of F is defined by h(F ) = {p ∈ βN : p ⊂ F }. Then h(F ) is a nonempty closed subset of βN and F ∈ F if and only if F ∩ h(F ) 6= ∅. Conversely, for a nonempty closed subset Z of βN, the kernel of Z is defined by k(Z) = {F ⊂ N : F ∩ Z 6= ∅}. Then k(Z) is a filterdual, h(k(Z)) = Z and k(h(F )) = F . In this means, we obtain a one-to-one corresponding between the set of filterduals on N and the set of nonempty closed subsets of βN ([10, 15]). Lemma 3.3. [15, 19] We have the following correspondences. (1) (2) (3) (4) h(Fps ) = K(βN). h(Fip ) = {p ∈ βN : p is an idempotent}. h(Fcen ) = {p ∈ βN : p is a minimal idempotent}. S h(Fpubd ) = {Supp(µ) : µ ∈ M}, where M is the set of all N-invariant probability measure on βN. Lemma 3.4. Let F be a filterdual. Then F is translation + invariant if and only if h(F ) is a closed left ideal of βN. Proof. Assume that F is translation + invariant. In order to show that h(F ) is a closed left ideal, it suffices to show that m + h(F ) ⊂ h(F ) for every m ∈ N. Let m ∈ N, p ∈ h(F ) and F ∈ m + p. Then m ∈ {n ∈ N : −n+F ∈ p} and −m+F ∈ p ⊂ F . Since F is translation DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 9 + invariant, m + (−m + F ) ⊂ F , then F ∈ F and m + p ⊂ F , i.e., m + p ∈ h(F ). Conversely, assume that h(F ) is a closed left ideal of βN. Let F ∈ F and n ∈ N, we want to show that n + F ∈ F . By Lemma 2.2, there exists some p ∈ h(F ) with F ∈ p. Clearly, n ∈ {m ∈ N : −m+(n+F ) ∈ p}, so n + F ∈ n + p ∈ h(F ) and n + F ∈ F . Lemma 3.5. Let F be a filterdual and bF = F . Then h(F ) is a closed two sided ideal of βN. Proof. Since bF is translation + invariant, by Lemma 3.4, h(F ) is a closed left ideal of βN. Then it suffices to show that h(F ) is also a right ideal. Let p ∈ h(F ), q ∈ βN and A ∈ p + q, we need to show that A ∈ F . Let F = {n ∈ N : −n T + A ∈ q}. Then F ∈ p ⊂ F . For every finite subset E of F , n∈E (−n + A) ∈ q is not empty, choose nE ∈ T n∈E (−n + A), then nE + E ⊂ A. This implies A ∈ bF = F . Let F be a filterdual, we call F ⊂ N an essential F -set if there is an idempotent p ∈ h(F ) such that F ∈ p. Denote by Fe the collection of all essential F -sets. Then Fe is also a filterdual since e = {p ∈ βN : p is an idempotent in h(F )}. h(F) We have the following observations: Let F be a subset of N. Then (1) F is an IP set if and only if it is an essential bFip -set. (2) F is a quasi-central set if and only if it is an essential Fps -set. (3) F is a D-set if and only if it is an essential Fpubd -set. (4) F is a C-set if and only if it is an essential J -set, where J is the collection of all J-sets. Theorem 3.6. Let F be a translation invariant filterdual and {xn }∞ n=1 be a sequence in N. If F S({xn }∞ ) ∈ F , then for every m ∈ N, n=1 ∞ F S({xn }n=m ) is an essential F -set. Proof. We first prove the following Tclaim. Claim: For each m ∈ N, h(F ) F S({xn }∞ n=m ) 6= ∅. Proof of the Claim: Clearly, the claim holds for m = 1. Now assume that m ≥ 2, then [ m−1 F S({xn }∞ F S({xn }∞ n=1 ) =F S({xn }n=1 ) n=m ) [ m−1 t + F S({xn }∞ n=m ) : t ∈ F S({xn }n=1 ) . Since F is translation invariant, p cannot be a principle ultrafilter, ∞ then the finite set F S({xn }m−1 n=1 ) is not in p. If F S({xn }n=m ) ∈ p, then 10 J. LI the claim holds. Now assume that we have some t ∈ F S({xn }m−1 n=1 ) ∞ ∞ such that t + F S({xn }n=m ) ∈ p. Choose q ∈ F S({xn }n=m ) such that t + q = p. For every F ∈ q, t ∈ {n ∈ N : −n + (t + F ) ∈ q}, so t + F ∈ p ⊂ F . Since F is translation invariant, we have F ∈ F and q ∈ h(F ). This ends the proof of the claim. T ∞ By Lemma 3.2 ∞ m=1 F S({xn }n=m ) is a closed subsemigroup of βN, and by Lemma 3.4 h(F ) is a closed left ideal of βN. Then by the above T T∞ ∞ claim we have h(F ) m=1 F S({xn }n=m ) is a nonempty subsemigroup of βN. By the well known Theorem, there exists some T T∞Ellis-Namakura ∞ idempotent in h(F ) m=1 F S({xn }n=m ). Thus for every m ∈ N, F S({xn }∞ ) is an essential F -sets. n=m For convenience, we also consider βZ+ the Stone-Čech compactification ofSZ+ . There is a nature imbedding map i : βN → βZ+ by i(p) = p {A ∪ {0} : A ∈ p}. Then we can regard βN as a subset of βZ+ and βZ+ = βN ∪ {0}. The advantage of βZ+ is that it contains the identity element 0, but we don’t want to take 0 into account when considering the multiplication. 4. Relationships between algebra properties of βN and dynamical properties A topological dynamical system (or just system) is a pair (X, T ), where X is a nonempty compact Hausdorff space and T is a continuous map from X to itself. When X is metrizable or T is a homeomorphism, we call (X, T ) a metrizable or invertible dynamical system respectively. Let (X, T ) be a dynamical system and x ∈ X, denote the orbit of x by Orb(x, T ) = {T n x : n ∈ Z+ }. Let ω(x, T ) be the ω-limit set of x, i.e., ω(x, T ) is the limit set of Orb(x, T ). A point x ∈ X is called a recurrent point if x ∈ ω(x, T ). We call a system (X, T ) is minimal if it contains no proper subsystems, and x ∈ X is a minimal point if it belongs to some minimal subsystem of X. A factor map π : (X, T ) → (Y, S) is a continuous surjective map from X to Y such that S ◦ π = π ◦ T . In this situation (X, T ) is said to be an extension of (Y, S) and (Y, S) the factor of (X, T ). Let F be a family and (X, T ) be a system, a point x ∈ X is called an F -recurrent point if for every open neighborhood U of x the entering time set N(x, U) = {n ∈ N : T n x ∈ U} ∈ F . If x is F -recurrent, then so is T x. Let π : (X, T ) → (Y, S) be a factor map, if x ∈ X is F -recurrent, then so is π(x). It is well known that x is recurrent if and only if it is Fip -recurrent and x is a minimal point if and only if DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 11 it is Fs -recurrent. If F is a filter, then x is F -recurrent if and only if F -lim T n x = x. Now we generalize the notion of ω-limit set. Let F be a family, (X, T ) be a dynamical system and x ∈ X, a point y ∈ X is called an F -ω-limit point of x if for every neighborhood U of y the entering time set N(x, U) ∈ F . Denote by ωF (x, T ) the set of all F -ω-limit points. Then x is F -recurrent if and only if x ∈ ωF (x, T ). An invariant measure for a dynamical system (X, T ) is a regular Borel probability measure µ on X such that µ(T −1A) = µ(A) for all Borel subsets A of X. Lemma 4.1. Let (X, T ) be a dynamical system and x ∈ X. If x is a recurrent point with Orb(x, T ) = X, then (1) x is Fps -recurrent if and only if (X, T ) has dense minimal points ([24]). (2) x is Fpubd -recurrent if and only if for every open neighborhood U of x there exists an invariant measure µ on (X, T ) such that µ(U) > 0 ([23, 4]). Lemma 4.2. Let F be a family and p ∈ βN. (1) If p is an idempotent and p ⊂ F , then p is F -recurrent in (βZ+ , λ1 ). (2) If p is F -recurrent in (βZ+ , λ1 ), then p ⊂ bF . Proof. (1) For every neighborhood U of p, there exists some F ∈ p such that F ⊂ U. Then N(p, F ) = {n ∈ N : (λ1 )n p ∈ F } = {n ∈ N : n + p ∈ F } = {n ∈ N : −n + F ∈ p}. Since F ∈ p = p + p, then {n ∈ N : −n + F ∈ p} ∈ p. Thus N(p, F ) ∈ F and p is F -recurrent. (2) For every F ∈ p, F is an open neighborhood of p and N(0, F ) = F . Let F ′ = N(p, F ). Since p is F -recurrent, F ′ ∈ F . For every finite subset W of F ′ , by the continuity of λ1 , there exists an open neighborhood U of p such that (λ1 )n U ⊂ F for every n ∈ W . Since p ∈ Orb(0, λ1), there exists some m ∈ Z+ such that (λ1 )m 0 ∈ U. Then m + W ⊂ N(0, F ). Thus, F ∈ bF . Let (X, T ) be a dynamical system. Then (X X , T ) also forms a dynamical system, where X X is endowed with its compact, pointwise convergence topology and T acts on X X as composition. The enveloping semigroup of (X, T ), denoted by E(X, T ), is defined as the closure of the set {T n : n ∈ Z+ } in X X . From the algebraic point of view, E(X, T ) is a compact Hausdorff right topological semigroup. On the other hand, (E(X, T ), T ) is a subsystem of (X X , T ). Those two structures are closely related. A 12 J. LI subset L ⊂ E(X, T ) is a closed left ideal of E(X, T ) if and only if (L, T ) is a subsystem of (E(X, T ), T ), and L is a minimal left ideal of E(X, T ) if and only if (L, T ) is a minimal subsystem of (E(X, T ), T ). If π : (X, T ) → (Y, S) is a factor map, then there is a unique continuous semigroup homomorphism π̃ : E(X, T ) → E(Y, S) such that π(px) = π̃(p)π(x). Let (X, T ) be a dynamical system and I be any nonempty set. Let I X be the product space and define T (I) : X I → X I by T (I) ((xi )i∈I ) = (T xi )i∈I . Then there is a natural isomorphism between E(X, T ) and E(X I , T (I) ). For convenience, we regard E(X, T ) acting on the factors of (X, T ) and the product systems of (X, T ). For each x ∈ X, there is a canonical factor map ϕx : E(X, T ) → (Orb(x, T ), T ), q 7→ qx. Let (X, T ) be a dynamical system. Z+ acts on X as Φ : Z+ × X → X, (n, x) 7→ T n x. Since βZ+ is the Stone-Čech compactification of Z+ , we can extend Φ to βZ+ × X → X, (p, x) 7→ px. For each x ∈ X, the map Φx : (βZ+ , λ1 ) → (Orb(x, T ), T ), p 7→ px is a factor map and Φx (βN∗ ) = ω(x, T ), where βN∗ = βN \ N. Lemma 4.3. Let (X, T ) be a system, x ∈ X and p ∈ βN. Then px = p - lim T n x. Proof. Clearly, the result holds for principle ultrafilters. Now we assume that p is a non-principle ultrafilter. Consider the factor map Φx : (βZ+ , λ1 ) → (Orb(x, T ), T ), p 7→ px. For every neighborhood U of px, let V = Φ−1 x (U), then V is a neighborhood p. There exists a subset F of N such that p ∈ F ⊂ V . Then F ⊂ N(0, V ) ⊂ N(x, U). Thus, N(x, U) ∈ p. We can also extend Ψ : Z+ → X X , n 7→ T n to βZ+ → E(X, T ). It is easy to see that Ψ is a semigroup homomorphism and Ψ : (βZ+ , λ1 ) → E(X, T ) is also a factor map. For every x ∈ X, Φx and ϕx ◦ Ψ agree on Z+ which is dense in βZ+ , then Φx = ϕx ◦ Ψ, i.e., the following diagram commutes. Ψ / (E(X, T ), T ) (βZ+ , λ1 ) Φx ♠♠ ♠♠♠ ♠ ♠ ♠♠ v ♠♠ ϕx ♠ (Orb(x, T ), T ) DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 13 Before continuing discussion, we need some preparation about symbolic dynamics. Let Σ2 = {0, 1}Z+ and σ : Σ2 → Σ2 by the shift map, i.e. the map (x(0), x(1), x(2), . . .) 7→ (x(1), x(2), x(3), . . .). Let [i0 i1 . . . in ] = {x ∈ Σ2 : x(0) = i0 , x(1) = i1 , · · · , x(n) = in } for ij ∈ 0, 1 and j = 0, 1, . . . , n. For any F ⊂ Z+ , we denote 1F be the indicator function from Z+ to {0, 1}, i.e., 1F (n) = 1 if n ∈ F and 1F (n) = 0 if n 6∈ F . In a natural way, each indicator function can be regarded as an element of Σ2 . It should be noticed that the enveloping semigroup of ({0, 1}Z+ , σ) is topologically and algebraically isomorphic to βZ+ ([10, 15]). Similarly, we can define two sided symbolic dynamics ({0, 1}Z, σ). Theorem 4.4. Let F be a filterdual. Suppose that h(F ) is a subsemigroup of βN. Let (X, T ) be a system and x ∈ X. Then the following conditions are equivalent: (1) x ∈ X is an F -recurrent point; (2) there exists an idempotent u ∈ h(F ) such that ux = x; (3) there exists an F -recurrent idempotent v ∈ E(X, T ) such that vx = x; (4) x is an Fe-recurrent point, where Fe is the collection of all essential F -sets. Proof. (1) ⇒ (2) Let A = {N(x, U) : U is an open neighborhood of x}. Then A ⊂ F and the intersection of any finite elements is also in A. By Lemma 2.2 there exists some p ∈ h(F ) such that A ⊂ p, thus px = x. Let L = {q ∈ βN : qx = x}. Then L is a closed subsemigroup of βN and so is L ∩ h(F ) since p ∈ L ∩ h(F ). By Ellis-Namakura Theorem there exists an idempotent u ∈ L ∩ h(F ). (2) ⇒ (3) Let v = Ψ(u). Since u is F -recurrent, v is also F -recurrent. Since Ψ is a semigroup homomorphism, vv = Ψ(u)Ψ(u) = Ψ(uu) = Ψ(u) = v. By Φx = ϕx ◦ Ψ, x = ux = Φx (u) = ϕx (Ψ(u)) = ϕx (v) = vx. (2) ⇒ (4), (3) ⇒ (1) and (4) ⇒ (1) are obvious. Proposition 4.5. Let F be a filterdual. Suppose that h(F ) is a subsemigroup of βN. Let π : (X, T ) → (Y, S) is a factor map. If y ∈ Y is an F -recurrent point, then there is an F -recurrent point x in π −1 (y). Proof. By Theorem 4.4 there exists an idempotent u ∈ h(F ) such that uy = y. Choose z ∈ π −1 (y) and let x = uz. Then π(x) = π(uz) = uπ(z) = uy = y and ux = uuz = uz = x, so x is F -recurrent and x ∈ π −1 (y). 14 J. LI Remark 4.6. Recall that a point x ∈ X is a minimal point if and only if it is Fs -recurrent. Unfortunately, Fs is not a filterdual. Can we use some filterdual instead of Fs to characterize minimal points? Intuitively, Fcen may be a good choice. But this is not true, it is shown in [25] that there exists an Fcen -recurrent point which is not a minimal point. Let (X, T ) be a system and x, y ∈ X. We call x, y are proximal if there exists some point z ∈ X such that (z, z) ∈ ω((x, y), T × T ). Proposition 4.7 ([10, 12, 5]). Let (X, T ) be a system and x, y ∈ X. Then the following conditions are equivalent: (1) x, y are proximal and y is a minimal point; (2) there exists a minimal idempotent u ∈ βN such that ux = uy = y; (3) there exists a minimal idempotent v ∈ E(X, T ) such that vx = vy = y; (4) (y, y) ∈ ωFcen ((x, y), T × T ). Let (X, T ) be a system and x, y ∈ X. We call x is strongly proximal to y if (y, y) ∈ ω((x, y), T × T ). It is easy to see that if y is a minimal point then x, y are proximal if and only if x is strongly proximal to y. Lemma 4.8. Let (X, T ) be a dynamical system and x, y ∈ X. Then the following conditions are equivalent: (1) x is strongly proximal to y; (2) (y, y) ∈ ωFip ((x, y), T × T ); (3) for every n ∈ N, x is strongly proximal to y in (X, T n ). Proof. (2)⇒(1) and (3)⇒(1) are obvious. (2)⇒(3) follows from the fact that if F is an IP set then for every n ∈ N the set {m ∈ N : mn ∈ F } is also an IP set. (1)⇒(2) Consider the factor map Φ(x,y) : (βZ+ , λ1 ) → (Orb((x, y), T × T ), T × T ), q 7→ q(x, y). T Let L = {p ∈ βN : p(x, y) = (y, y)} = Φ−1 βN. Then L is not (x,y) (y, y) empty closed subset of βN, since (y, y) ∈ ω((x, y), T ×T ). We show that L is subsemigroup of βN. Let p, q ∈ L. Then p(x, y) = (px, py) = (y, y) and q(x, y) = (qx, qy) = (y, y), so pq(x, y) = (pqx, pqy) = (py, py) = (y, y). By Ellis-Namakura Theorem there exists an idempotent p in L. Then by Lemma 4.3 and p ⊂ Fip one has (y, y) ∈ ωFip ((x, y), T ×T ). Let F be a family, (X, T ) be a system and x, y ∈ X, we call x is F -strongly proximal to y if (y, y) ∈ ωF ((x, y), T × T ) ([1]). DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 15 Theorem 4.9. Let F be a filterdual. Suppose that h(F ) is a subsemigroup of βN. Let (X, T ) be a system and x, y ∈ X. Then the following conditions are equivalent: (1) x is F -strongly proximal to y; (2) there exists an idempotent u ∈ h(F ) such that ux = uy = y; (3) there exists an F -recurrent idempotent v ∈ E(X, T ) such that vx = vy = y; (4) x is Fe-strongly proximal to y. Proof. (1) ⇒ (2) Let A = {N((x, y), U × U) : U is an open neighborhood of y}. By the definition of F -strong-proximity, we have A ⊂ F and the intersection of finite elements in A is also in A. Then by Lemma 2.2 there exists some p ∈ h(F ) such that A ⊂ p, so p(x, y) = (y, y). Let L = {q ∈ βN : qx = qy = y}. Then L ∩ h(F ) is a nonempty closed subsemigroup of βN. By Ellis-Namakura Theorem there exists an idempotent u ∈ L ∩ h(F ). (2) ⇒ (3) Let v = Ψ(u). Since u is F -recurrent, v is also F -recurrent. Then by Φ(x,y) = ϕ(x,y) ◦ Ψ we have vx = vy = y. (3) ⇒ (2) By Theorem 4.4 there exists an idempotent u ∈ h(F ) such that v = uv = Ψ(u). Then by Φ(x,y) = ϕ(x,y) ◦ Ψ we have ux = uy = y. (2) ⇒ (4) Since u(x, y) = (y, y) and u is an idempotent in h(F ), by Lemma 4.3 (y, y) ∈ ωFe ((x, y), T × T ). (4) ⇒ (1) is obvious. Proposition 4.10. Let F be a filterdual. Suppose that bF = F . Let (X, T ) be a system and x, y ∈ X. Then x is F -strongly proximal to y if and only if y is an F -recurrent point and x is strongly proximal to y. Proof. By the definition, if x is F -strongly proximal to y, then y is an F -recurrent point and x is strongly proximal to y. Conversely, assume that y is an F -recurrent point and x is strongly proximal to y. Consider the factor map Φ(x,y) : (βZ+ , λ1 ) → (Orb((x, y), T × T ), T × T ), p 7→ p(x, y). Since (y, y) ∈ Orb((x, y), T × T ) and (y, y) is F -recurrent, by Proposition 4.5 there exists an F -recurrent point q in βN with q(x, y) = (y, y). By Lemma 4.2 we have q ⊂ bF = F , then (y, y) ∈ ωF ((x, y), T ×T ). Now we can set up a general correspondence between essential F -sets and the sets defined by F -strong proximity. 16 J. LI Theorem 4.11. Let F be a filterdual. Suppose that h(F ) is a subsemigroup of βN. Then a subset F of N is an essential F -set if and only if there exists a dynamical system (X, T ), a pair of points x, y ∈ X where x is F -strongly proximal to y, and an open neighborhood U of y such that F = N(x, U). Proof. The sufficiency follows from Theorem 4.9 and N((x, y), U ×U) ⊂ N(x, U). Now we show the necessity. If F is an essential F -set, there exists an idempotent u ∈ h(F ) such that F ∈ u. Let x = 1F ∈ {0, 1}Z+ and y = ux. Then ux = y = y, so x is F -strongly proximal to y. Clearly, N(x, [1]) = F . Then it suffices to show that y ∈ [1]. If not, then y ∈ [0]. Thus, N(x, [0]) ∈ p and N(x, [0]) ∩ N(x, [1]) 6= ∅, This is a contradiction. Remark 4.12. (1) In the proof of Theorem 4.11, if we use {0, 1}Z instead of {0, 1}Z+ , then it shows that every essential F -set can be realized by an invertible metrizable system. (2) Since Fps and Fpubd are filterduals, and bFps = Fps , bFpubd = Fpubd, Theorem 1.5 and Theorem 1.6 are special cases of Theorem 4.11. We now give a combinatorial characterization of essential F -set. Proposition 4.13. Let F be a filterdual. Suppose that h(F ) is a subsemigroup of βN. Then a subset F of N is an essential F -set if and only if there is a decreasing sequence {Cn }∞ n=1 of subsets of F such that for every n ∈ N, Cn ∈ F and for every r ∈ Cn there exists m ∈ N such that r + Cm ⊂ Cn . Proof. If F is an essential F -set, there exists an idempotent u ∈ h(F ) such that F ∈ u. Let x = 1F ∈ {0, 1}Z+ and y = ux. Then u(x, y) = (y, y), y ∈ [1] and N(x, [1]) = F . For each n ∈ N, let Un = [y(0)y(1) . . . y(n)] and Cn = N((x, y), Un × Un ), then by Theorem 4.9 each Cn is an essential F -set. For every r ∈ Cn , we have (σ × σ)r (y, y) ∈ Un × Un . By the continuity of σ, there exists m ∈ N such that (σ × σ)r (Um × Um ) ⊂ Un × Un , then r + Cm ⊂ Cn . Conversely, assume that there is a decreasing sequence {Cn }∞ n=1 satisfying the condition. By LemmaT2.2 there exists some p ∈ h(F ) such that {Cn : n ∈ N} ⊂ p. Let L = ∞ n=1 Cn . By Lemma 3.2 L is a closed subsemigroup of βN. Then p ∈ L ∩ h(F ) and L ∩ h(F ) is nonempty closed subsemigroup of βN. By Ellis-Namakura Theorem there exists an idempotent in L ∩ h(F ). Thus, each Cn is an essential F -set. In particular, F is an essential F -set. Corollary 4.14. Let p be an idempotent βN and F ⊂ N. Then F ∈ p if and only if there is a decreasing sequence {Cn }∞ n=1 of subsets of F DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 17 such that for every n ∈ N, Cn ∈ p and for every r ∈ Cn there exists m ∈ N such that r + Cm ⊂ Cn . 5. The set’s forcing In this section, we discuss the set’s forcing. This terminology was first introduced in [7], the idea goes back at least to [11] and [15]. We say that a subset F of N forces F -recurrence if for every dynamical system (X, T ) and x ∈ X there exists some F -recurrent point in T F x, where T F x = {T n x : n ∈ F }. In [11] and [15], the authors call a subset F of N is big if there exists a minimal point in Orb(x, σ) ∩ [1], where x = 1F ∈ Σ. Proposition 5.1 ([15, 7]). Let F be a subset of N. Then the following conditions are equivalent: (1) F is big; (2) F is piecewise syndetic; (3) F forces Fs -recurrence; (4) there exists a minimal left ideal L of βN such that F ∩ L 6= ∅. Let F be a family, denote by F orce(F ) the collection of all sets that force F -recurrence. Clearly, F orce(F ) is a family. It is easy to see that F orce(F ) is not empty if and only if there exists some F -recurrent point in (βZ+ , λ1 ). Theorem 5.2. Let F be a family and F be a subset of N. Then F ∈ F orce(F ) if and only if there exists an F -recurrent point p ∈ βN such that F ∈ p. Proof. Let F ∈ F orce(F ). Consider the system (βZ+ , λ1 ) and 0 ∈ βZ+ , since F forces F -recurrence, there exists an F -recurrent point p ∈ (λ1 )F 0 = {(λ1 )n 0 : n ∈ F } = F . Thus, F ∈ p. Conversely, assume that there exists an F -recurrent point p ∈ βN such that F ∈ p. For every dynamical system (X, T ) and x ∈ X, consider the factor map Φx : (βZ+ , λ1 ) → (Orb(x, T ), T ). Let y = px. Then y is F -recurrent. Then it suffices to show that y ∈ T F x. For every open neighborhood U of y, N(x, U) ∈ p. Since F ∈ p, N(x, U)∩F 6= ∅, thus y ∈ T F x. Corollary 5.3. Let F be a family. Then [ h(F orce(F )) = {βZ+ + p : p is an F -recurrent point}. Proposition 5.4. Let F be a family. If F orce(F ) is not empty, then F orce(F ) is a filterdual and F orce(F ) = b(F orce(F )) ⊂ bF . 18 J. LI Proof. Let F ∈ F orce(F ) and F = F1 ∪F2 . If both F1 and F2 are not in F orce(F ), then there exist two dynamical systems (X, T ) , (Y, S) and two points x ∈ X, y ∈ Y such that both T F1 x and S F2 x contain no F recurrent points. Consider the system (X×Y, T ×S) and (x, y) ∈ X×Y . Since F forces F -recurrence, there exists an F -recurrent point [ (z1 , z2 ) ∈ (T × S)F (x, y) = (T × S)F1 (x, y) (T × S)F2 (x, y). Without loss of generality, assume that (z1 , z2 ) ∈ (T × S)F1 (x, y). Then z1 ∈ T F1 x and z1 is F -recurrent, this is a contradiction. Thus, F orce(F ) is a filterdual. Let F ∈ b(F orce(F )). Then a sequence {an } in Z+ S there exists ′ and F ′ ∈ F orce(F ) such that ∞ (a + F ∩ [1, n]) ⊂ F . Let (X, T ) n=1 n be a dynamical system and x ∈ X. Since X is compact, there is a subnet {ani } of {an } such that lim T ani x = y. Since F forces F recurrence, there exists an F -recurrent point z ∈ T F ′ y. Then it suffices to show that z ∈ T F x. For every open neighborhood U of z, there exists k ∈ F ′ such that T k y ∈ U. By the continuity of T , choose an open neighborhood V of y such that T k V ⊂ U. Since lim T ani x = y and {ani } is a subnet of {an }, there exists some n ≥ k such that T an x ∈ V . Then an + k ∈ F and T an +k x ∈ U, so z ∈ T F x. Let F ∈ F orce(F ), we show that F ∈ bF . Let x = 1F ∈ {0, 1}Z+ . Since F forces F -recurrence, there exists an F -recurrent point y ∈ T F x. Clearly, y ∈ [1] and N(x, [1]) = F . Let N(y, [1]) = F ′ . Then F ′ ∈ F . For every finite subset W of F ′ , by the continuity of σ, there exists an open neighborhood U of y such that σ n (U) ⊂ [1] for every n ∈ W . Since y ∈ Orb(x, σ), choose m ∈ Z+ such that σ m x ∈ U, then m + W ⊂ N(x, [1]). So F ∈ bF . Theorem 5.5. Let F be a filterdual and F be a subset of N. Suppose that h(F ) is a subsemigroup of βN. Then the following conditions are equivalent: (1) F forces F -recurrence; (2) let x = 1F ∈ {0, 1}Z+ , there exists an F -recurrent point in T Orb(x, σ) [1]; (3) F is a block essential F -set, i.e., F ∈ bFe. Proof. (1) ⇒ (2) Let x = 1F ∈ {0, 1}Z+ . Since F forces F -recurrence, there exists an F -recurrent point y in σ F x ⊂ [1]. T (2) ⇒ (3) Choose an F -recurrent point y in Orb(x, σ) [1]. By e Theorem 4.4 y is also F-recurrent. Since N(x, [1]) = F and N(y, [1]) ∈ e F, by the continuity of σ we have F ∈ bFe. DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 19 (3) ⇒ (1) By Proposition 5.4, it suffices to show every essential F e Then there exists an idempotent set forces F -recurrence. Let F ∈ F. u ∈ h(F ) such that F ∈ u. Let (X, T ) be a dynamical system and x ∈ X. Let y = ux. Then uy = y, so y is F -recurrent. For every open neighborhood U of y, N(x, U) ∈ u. Since F ∈ u, F ∩ N(x, U) 6= ∅, thus y ∈ T F x. Corollary 5.6. Let F be a filterdual and F be a subset of N. Suppose that h(F ) is a subsemigroup of βN. Let (X, T ) be a dynamical system and x ∈ X. Then x is the unique F -recurrent point in (X, T ) if and only if for every y ∈ X, κ(bFe)-lim T n y = x. Proof. Since F is a filterdual, κ(bFe) is a filter. If x is the unique F -recurrent point, then by Theorem 5.5 for every y ∈ X and every F ∈ bFe we have x ∈ T F y, so κ(bFe)-lim T n y = x. Conversely, assume that there exists another F -recurrent point y ∈ X. Choose open subsets U, V of X such that x ∈ U, y ∈ V and U ∩ V = ∅. Then N(y, U) ∈ κ(bFe) and N(y, V ) ∈ Fe ⊂ bFe. Thus, N(y, U) ∩ N(y, V ) 6= ∅. This is a contradiction. fip , bFip = bF fip . Then a subset F of N Remark 5.7. (1) Since Fip = F forces recurrence if and only if F ∈ bFip ([7]). (2) It is shown in [27] that a subset F of N forces Fpubd -recurrence if ] and only if F ∈ Fpubd, i.e., bF pubd = Fpubd . For completeness, we include a proof. Let F ∈ Fpubd and x = 1F ∈ {0, 1}Z+ . By [12, Lemma 3.17], T there exists a σ-invariant measure µ such that µ(Orb(x, σ) [1]) > 0. By the ergodic decomposition theorem, choose an ergodic σ-invariant T measure ν such that ν(Orb(x, σ) [1]) > 0. Then pick a generic point T y in Orb(x, σ) [1] for ν and so y is Fpubd-recurrent ([12, pp. 62-64]). Thus, F forces Fpubd -recurrent. It is interesting that central sets also have some kind of forcing. Proposition 5.8. Let F be a subset of N. Then the following conditions are equivalent: (1) F is central; (2) let x = 1F ∈ {0, 1}Z+ , there exists some minimal point y ∈ Orb(x, σ) ∩ [1] such that x, y are proximal; (3) for every dynamical system (X, T ) and x ∈ X there exists some minimal point y ∈ T F x such that x, y are proximal. Proof. (2) ⇒ (1) follows from the definition of central sets and N(x, [1]) = F. 20 J. LI (3) ⇒ (2) follows from T F x ⊂ [1]. (1) ⇒ (3) If F is central, then there exists a minimal idempotent u ∈ βN such that F ∈ u. Let (X, T ) be a dynamical system and x ∈ X. Let y = ux. Then ux = uy = y, so y is a minimal point and x, y are proximal. Thus it suffices to show that y ∈ T F x. For every open neighborhood U of y, N(x, U) ∈ u. Since F ∈ u, F ∩ N(x, U) 6= ∅, so y ∈ T F x. We say a subset F of N forces F -strong proximity if for every dynamical system (X, T ) and x ∈ X there exists some point y in T F x such that x is F -strongly proximal to y. Proposition 5.9. Let F be a filterdual. Suppose that h(F ) be a subsemigroup of βN. Let F be a subset of N. Then the following conditions are equivalent: (1) F is an essential F -set; (2) let x = 1F ∈ {0, 1}Z+ , there exists some point y ∈ Orb(x, σ)∩[1] such that x is F -strongly proximal to y; (3) F forces F -strong proximity. Proof. (2) ⇒ (1) follows from Theorem 4.11 and N(x, [1]) = F . (3) ⇒ (2) follows from T F x ⊂ [1]. (1) ⇒ (3) If F is an essential F -set, then there exists an idempotent u ∈ h(F ) such that F ∈ u. Let (X, T ) be a dynamical system and x ∈ X. Let y = ux. Then ux = uy = y and by Theorem 4.9 x is F -strongly proximal to y. Thus it suffices to show that y ∈ T F x. For every open neighborhood U of y, N(x, U) ∈ u. Since F ∈ u, F ∩ N(x, U) 6= ∅, so y ∈ T F x. 6. Multiplication in N and βN In this section, we consider both addition and multiplication in N and βN. For n ∈ N and F ⊂ N, let nF = {nm : m ∈ F } and n−1 F = {m ∈ N : nm ∈ F }. For p, q ∈ βN, the multiplication p · q in βN is {A ⊂ N : {n ∈ N : n−1 A ∈ q} ∈ p}. A family F is called multiplication invariant if for each n ∈ N and F ∈ F one has nF ∈ F . It is easy to see that Fip , Fs and Fpubd are multiplication invariant. Similarly to Lemma 3.4, we have Lemma 6.1. Let F be a filterdual. Then F is multiplication invariant if and only if h(F ) is a left ideal of (βN, ·). DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 21 Proposition 6.2 ([12, 6]). Let F be a subset of N. If F is a central set, then for each n ∈ N both nF and n−1 F are also central sets. The main purpose of this section is to extend Proposition 6.2 to more general settings. In particular, similar results hold for quasi-center sets and D-sets. Theorem 6.3. Let F be a filterdual and F be a subset of N. Suppose that F is multiplication invariant and h(F ) is a subsemigroup of (βN, +). If F is an essential F -set, then for each n ∈ N, nF is also an essential F -set. Proof. Let x = 1F ∈ {0, 1}Z+ . Then by Proposition 5.9 there exists a point y ∈ σ F x ⊂ [1] such that F -lim(σ × σ)m (x, y) = (y, y). Fix n ∈ N and let Y = {1, 2, . . . , n} endowing with discrete topology and X = {0, 1}Z+ × Y . Define T : X → X by T (z, i) = (z, i + 1) for i ≤ n − 1 and T (z, n) = (σz, 1). For every neighborhood U of y, we have N((x, 1, y, 1), U × {1} × U × {1}) = nN((x, y), U × U). Since F is multiplication invariant, F -lim(T ×T )m (x, 1, y, 1) = (y, 1, y, 1). Thus, nF = N((x, 1), [1] × {1}) is also an essential F -set. We call F -recurrence is iteratively invariant if for every dynamical system (X, T ) and every F -recurrent point x in (X, T ), x is also an F -recurrent point in (X, T n ) for each n ∈ N. It is well known that Fip -recurrence and Fs -recurrence are iteratively invariant. We show that Theorem 6.4. Let F be a filterdual and F be a subset of N. Suppose that bF = F and F -recurrence is iteratively invariant. If F is an essential F -set, then for each n ∈ N, n−1 F is also an essential F -set. Proof. Let x = 1F ∈ {0, 1}Z+ . Then by Proposition 5.9 there exists an F -recurrent point y ∈ σ F x ⊂ [1] such that x is strongly proximal to y. For each n ∈ N, since F -recurrence is iteratively invariant, y is also an F -recurrent point in ({0, 1}Z+ , σ n ). By Lemma 4.8, x also is strongly proximal to y in ({0, 1}Z+ , σ n ). Then by Proposition 4.10 and Theorem 4.9 n−1 F = {m ∈ N : (σ n )m x ∈ [1]} is an essential F -set. A system (X, T ) is called topologically transitive if for every two nonempty open subset U, V of X there exists some n ∈ N such that T n U ∩ V 6= ∅. A point x ∈ X is called a transitive point if the orbit of x is dense in X. The system (X, T ) is called point transitive if there exists some transitive point in X. In general, there is no implicational relation between topological transitivity and point transitivity. For 22 J. LI example, (βZ+ , λ1 ) is point transitive but not topologically transitive. The system (X, T ) is called recurrent transitive if there exists some recurrent transitive point, i.e., there exists x ∈ X such that ω-limit set of x is X. It is easy to see that every recurrent point transitive system is topologically transitive. The following result is a “folk” result, for similar results, see [2] for example. Lemma 6.5. Let (X, T ) be a recurrent transitive system. Then for every n ∈ N there is k ∈ N with k|n and a decomposition X = X0 ∪ X1 ∪ · · · ∪ Xk−1 satisfying (1) Xi 6= Xj , 0 ≤ i < j ≤ k − 1, (2) T Xi = Xi+1 (mod k) , (3) (Xi , T n ) is recurrent transitive, i = 0, . . . , k − 1, (4) the interior of Xi is dense in Xi , i = 0, . . . , k − 1. Proof. Let x ∈ X with ω(x, T ) = X. Let Yi = Orb(T i x, T n ) for i = 0, 1, . . . , n − 1. Then X = Y0 ∪ Y1 ∪ · · · ∪ Yn−1 and T Yi = Yi+1 (mod n) . Since x is recurrent in (X, T ), T i x is also recurrent in (X, T n ). Then (Yi , T n ) is recurrent transitive for i = 0, 1, . . . , n − 1. Let k be the smallest positive integer such that T k Y0 = Y0 . Let Xi = Yi for i = 0, 1, . . . , k − 1. Now we show that those Xi satisfy the requirement. Clearly, k ≤ n. Let n = lk + r with l > 0 and 0 ≤ r < k. Then X0 = T n (X0 ) = T r (T lk X0 ) = T r (X0 ), by the minimality of k, we have r = 0, so k|n. If there exist 0 ≤ i < j ≤ k − 1 such that Xi = Xj , then T j−i X0 = j−i T (T n X0 ) = T n−i (T j X0 ) = T n−i (T i X0 ) = T n X0 = X0 . This contradicts the minimality of k. So Xi 6= Xj for 0 ≤ i < j ≤ k − 1. For 0 ≤ i 6= j ≤ k − 1 let Zij = Xi ∩ Xj , then Zij is a T n -invariant closed subset of Xi . Since (Xi , T n ) is topologically transitive, Zij either equals to Xi or is nowhere dense in Xi . If Zij = Xi , then Xi ⊂ Xj . Without loss of generality, assume i < j, then X0 = T k−i Xi ⊂ T k−i Xj = Xj−i . Thus, X0 ⊂ Xj−i ⊂ X2(j−i) (mod k) ⊂ · · · ⊂ Xk(j−i) (mod k) = X0 . This contradicts the minimality of k. So Zij is S nowhere dense in Xi . Now fix i ∈ {0, 1, . . . , k − 1} and let Zi = j6=i Zij . Then Zi is also nowhere dense in Xi . The boundary of Xi in X is \[ [ \ ∂Xi = Xi ∩ (X \ Xi ) ⊂ Xi ( Xj ) = (Xi Xj ) = Zi . By Xi = int(Xi ) S j6=i j6=i Zi , we have the interior of Xi is dense in Xi . DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 23 Lemma 6.6. Fps -recurrence and Fpubd-recurrence are iteratively invariant. Proof. Let (X, T ) be a dynamical system and x ∈ X be an Fps recurrent point. Without loss of generality, assume that Orb(x, T ) = X. By Lemma 4.1, (X, T ) has dense minimal points. For every n ∈ N, (X, T n ) also has dense minimal points. By Lemma 6.5, the interior of Orb(x, T n ) is dense in Orb(x, T n ), so (Orb(x, T n ), T n ) also has dense minimal points. Thus x is Fps -recurrent in (X, T n ). Let (X, T ) be a dynamical system and x ∈ X be an Fpubd-recurrent point. Without loss of generality, assume that Orb(x, T ) = X. By Lemma 4.1 and (X, T ) is transitive, for every nonempty open subset U of X there exists a T -invariant measure µ on X such that µ(U) > 0. For every n ∈ N, by Lemma 6.5, the interior of Orb(x, T n ) is dense in Orb(x, T n ). Then for every nonempty open subset V of Orb(x, T n ) there exists an open subset U of X such that U ⊂ V . So there exists a T -invariant measure µ on X such that µ(U) > 0. Clearly, µ is also a T n -invariant measure on X. Define a measure ν on Orb(x, T n ) by ν(A) = µ(A)/µ(Orb(x, T n)) for every Borel subset A of Orb(x, T n ). Then ν is a T n -invariant measure on Orb(x, T n ) with ν(V ) > 0. Thus x is Fpubd-recurrent in (X, T n ). Proposition 6.7. Let F be a subset of N and n ∈ N. (1) If F is a quasi-central set, then both nF and n−1 F are also quasi-central sets. (2) If F is a D-set, then both nF and n−1 F are also D-sets Proof. It follows from Theorem 6.4, Theorem 6.3, Lemma 6.6 and the fact that Fps and Fpubd are multiplication invariant. 7. Dynamical characterization of C-sets In this section, we show the following dynamical characterization of C-sets. Theorem 7.1. Let F be a subset of N. Then F is a C-set if and only if there exists a dynamical system (X, T ), a pair of points x, y ∈ X where y is J -recurrent and x is strongly proximal to y, and an open neighborhood U of y such that N(x, U) = F . By Proposition 4.10 and Theorem 4.11, it suffices to show the following two lemmas. Lemma 7.2. J is a filterdual. 24 J. LI Lemma 7.3. J = bJ and it is multiplication invariant. Then h(J ) is a closed two sided ideal in (βN, +) and left ideal in (βN, ·). Proof of Lemma 7.2. Let F be a J-set and F = F1 ∪ F2 . Using an argument due to [21, Theorem 2.14], we first show the following claim. (1) (m) Claim: For every IP-system {sα = (sα , . . . , sα )} in Zm , there exists an i ∈ {1, 2}, r ∈ Z and α ∈ Pf (N) such that r̄ (m) + sα ∈ Fim . Proof of the Claim: For j = 1, 2, . . . , m, define fj : N → Z by P (j) (j) fj (n) = s{n} , then sα = n∈α fj (n) for α ∈ Pf (N). Pick by Hales-Jewett Theorem ([16]) some n ∈ N such that whenever the length n words over the alphabet {1, . . . , m} are 2-colored, there exists a variable word w(v) such that {w(j) : j = 1, . . . , m} is monochromatic. Let W be the set of length n words over {1, . . . , m}. For w = b1 b2 · · · bn ∈ W define gw : N → Z by for l ∈ Z+ and i = 1, 2, . . . , n, gw (ln + i) = fbi (ln + i). For l ∈ Z+ , let Hl = {ln + 1, ln + 2, . . . , ln + n}. P P (w) For every w ∈ W and α ∈ Pf (N), let hα = l∈α t∈Hl gw (t). Then (w) (hα ) = (hα : w ∈ W ) is an IP-system in Z|W | . Then there exists a (w) r ∈ Z and α ∈ Pf (N) such that r + hα ∈ F for every w ∈ W . Define (w) φ : W → {0, 1} by φ(w) = 1 if r + hα ∈ F1 . Pick a variable word w(v) such that {w(j) : j = 1, 2, . . . , m} is monochromatic with respect to φ. Without loss of generality assume that φ(w(j)) = 1 for j = 1, 2, . . . , k. Let w(v) = c1 c2 · · · cn where each ci ∈ {1, 2, . . . , m} ∪ {v}. Let A = {i ∈ {1, 2, . . . , n} : ci = v} = 6 ∅ and B = {1, 2, . . . , n} \ A. For l ∈ Z+ , let HlA = Hl ∩ (ln + A) and HlB = Hl ∩ (ln + B). For j = 1, 2, . . . , m, (w(j)) rewrite hα as XX XX XX gw(j) (t) = h(w(j)) = gw(j)(t) + gw(j) (t). α l∈α t∈Hl l∈α t∈H A l l∈α t∈H B l P P P Then t∈H A gw(j) (t) = t∈H A fj (t) and t∈H B gw(j)(t) does not del l S P lP pend on j. Let α′ = l∈α HlA and r ′ = r + l∈α t∈H B gw(j)(t). Then (w(j)) (j) l r + hα = r ′ + sα′ . So r̄ ′(m) + sα′ ∈ F1m . This ends the proof of the Claim. We now show that in the Claim we can pick r ∈ N instead of r ∈ Z. (1) (m) (0) For every IP-system {sα = (sα , . . . , sα )} in Zm , let sα = −|α| for (0) (1) (m) each α ∈ Pf (N) and {s′α = (sα , sα , . . . , sα )}. Applying the Claim to {s′α }, there exists an i ∈ {1, 2}, r ∈ Z and α ∈ Pf (N) such that (0) (0) r̄ (m+1) + s′α ∈ Fim+1 . Since r + sα ∈ Fi and sα is negative, r must be positive. DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 25 (1) (m) If both F1 and F2 are not J-sets, let {sα = (sα , . . . , sα )} and ′(1) ′(m′ ) ′ {sα = (sα , . . . , sα )} be witnesses to the fact that F1 and F2 are not (1) (m) ′(1) ′(m′ ) J-sets. Let s′′α = (sα , . . . , sα , sα , . . . , sα ). Applying the Claim to {s′′α }, we get a contradiction. Proof of Lemma 7.3. If F is a block J-set, exists a sequence S∞ then there ′ ′ {an } in Z+ and F ∈ J such that n=1 (an + F ∩ [1, n]) ⊂ F . For every IP-system {sα } in Zm , there exists r ∈ N and α ∈ Pf (N) such that r̄ (m) + sα ∈ F ′(m) . Choose n large enough such that r̄ (m) + sα ∈ (F ′ ∩ [1, n])(m) and let r ′ = r + an . Then r̄ ′(m) + sα ∈ F m . Hence, F is also a J-set. Let F be a J-set and n ∈ N, we want to show that nF is also a J-set. Let {sα } be an IP-system in Zm . Without loss of generality, assume that {sα } ⊂ nZm . Let s′α = n−1 sα . Then {s′α } is also an IP-system in Zm . Since F is a J-set, there exists r ∈ N and α ∈ Pf (N) such that r̄ (m) + s′α ∈ F (m) , then nr ¯ (m) + sα ∈ nF (m) . Hence, nF is also a J-set. Remark 7.4. It is shown in [17] that there exists a C-set with upper Banach density 0. Then there exists a dynamical system (X, T ) and x ∈ X such that x is J -recurrent but not Fpubd -recurrent. 8. Solvability of Rado systems in C-sets In order to show that Rado systems are solvable in C-sets, by the method developed in [12, pp.169–174], it suffices to show the following two results. Lemma 8.1. If F is a C-set, then for each n ∈ N, nF and n−1 F are also C-sets. Theorem 8.2. Let F be a C-set. Then for every m ∈ N and every IP-system {sα } in Zm there exists an IP-system {rα } in N and an (m) IP-subsystem {sφ(α) } such that for every α ∈ Pf (N), r̄α + sφ(α) ∈ F m . To discuss J -recurrence, we first introduce a new kind of dynamical system. Let (X, T ) be an invertible dynamical system, we say that (X, T ) satisfies the multiple IP-recurrence property if for every (1) (m) IP-system {sα = (sα , . . . , sα )} in Zm and every open subset U of X, there exists some α ∈ Pf (N) such that m \ i=1 (i) T −sα U 6= ∅. 26 J. LI Let (X, T ) be an invertible dynamical system, if it is a minimal system or there exists an invariant measure with full support, then it satisfies the multiple IP-recurrence property ([12, 13]). Lemma 8.3. Let (X, T ) be an invertible dynamical system and n ∈ N. Then the following conditions are equivalent: (1) (X, T ) satisfies the multiple IP-recurrence property; (1) (m) (2) for every IP-system {sα = (sα , . . . , sα )} in Zm , every open subset U of X and k ∈ N, there exists some α ∈ Pf (N) with min α > k such that ! m \ \ (i) T −sα U 6= ∅; U i=1 (3) (X, T n ) satisfies the multiple IP-recurrence property. Proof. (1)⇒(3) and (2)⇒ (1) are obvious. (m) (1) (1)⇒(2) Let {sα = (sα , . . . , sα )} be an IP-system Zm and k ∈ N. Define a homomorphism φ : Pf (N) → Pf (N) determined by φ({i}) = (0) {i + k} for any i ∈ N. Let sα = 0 for any α ∈ Pf (N). Then {s′α = (0) (1) (m) (sα , sφ(α) , . . . , sφ(α) )} is an IP-system in Zm+1 . Now (2) follows from applying to {s′α }. (3)⇒(1) Let {sα } be an IP-system in Zm . Without loss of generality, assume that {sα } ⊂ nZm . Let s′α = n−1 sα . Then {s′α } is also an IPsystem in Zm . Then (1) follows from applying to {s′α } in (X, T n ). Let {xα }α∈Pf (N) be a sequence in topological space X and x ∈ X, we say that xα → x as a Pf (N)-sequence if for every neighborhood U of x there exists αU ∈ Pf (N) such that xα ∈ U for all α > αU . If {xα } is a Pf (N)-sequence in a compact metric space, then there exists a Pf (N)-subsequence {xφ(α) } which converges as Pf (N)-sequence ([12, Theorem 8.14]). Proposition 8.4. Let (X, T ) be an invertible metrizable dynamical system. Then (X, T ) satisfies the multiple IP-recurrent property if and only if for every IP-system {sα } in Zm and every open subset U of X there exists x ∈ U and an IP-subsystem {sφ(α) } such that T for i = 1, . . . , m. (i) sφ(α) x→x Proof. The sufficiency is obvious. (1) (2) (m) We now show the necessity. Let {sα = (sα , sα , . . . , sα )} be an IP-system in Zm and U be an open subset of X. Let U0 = U. By DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 27 Lemma 8.3, there exists α1 ∈ Pf (N) such that ! m \ \ (i) U0 T −sα1 U0 6= ∅. i=1 Then choose an open subset U1 with U1 ⊂ U0 and diam(U1 ) < 1 such that m [ (i) T sα1 U1 ⊂ U0 . i=1 We now proceed inductively to define a sequence of open subset U1 , U2 , . . . , Un , . . . in X and a sequence α1 < α2 < · · · < αn < · · · in Pf (N) such that m [ (i) 1 Un+1 ⊂ Un , diam(Un ) < , T sαn Un ⊂ Un−1 . n i=1 T∞ Then there will be a unique point x in n=1 Un . Now set φ({n}) = αn for each n ∈ N. For every β = {r1 < r2 < · · · < rk } if min β > n + 1 then m m [ [ (i) (i) (i) (i) s T φ(β) Urk = T sαr1 T sαr2 · · · T sαrk Urk ⊂ Ur1 −1 ⊂ Un . i=1 i=1 (i) Hence, for i = 1, 2, . . . , m, T sφ(β) x ∈ Un if min β > n + 1. It follows (i) that T sφ(α) x → x for i = 1, 2, . . . , m. Theorem 8.5. Let (X, T ) be an invertible dynamical system and x ∈ X. Then x is J -recurrent if and only if (Orb(x, T ), T ) satisfies the multiple IP-recurrence property. Proof. Without loss of generality, assume that Orb(x, T ) = X. If x is J -recurrent, then for every open subset U of X there exists k ∈ N and an open neighborhood V of x such that T k V ⊂ U. Since x is J -recurrent, N(x, V ) is a J-set. Then for every IP-system {sα = (1) (m) (sα , . . . , sα )} in Zm there exists r ∈ N and α ∈ Pf (N) such that (i) (i) T r+sα x ∈ V for i = 1, . . . , m. Let y = T r+k x. Then T sα y = T (i) (i) −sα U. T k (T r+sα x) ∈ T k V ⊂ U for i = 1, . . . , m. So y ∈ m i=1 T Conversely, assume that (X, T ) satisfies the multiple IP-recurrence property. It is easy to see that x is recurrent. For every open neigh(1) (m) borhood U of x and every IP-system {sα = (sα , . . . , sα )} in Zm , T (i) −sα U 6= ∅. Choose there exists some α ∈ Pf (N) such that m i=1 T Tm −sα(i) (i) sα U, then T y ∈ U for i = 1, . . . , m. By the continuity y ∈ i=1 T (i) of T , choose an open neighborhood V of y such that T sα V ∈ U for 28 J. LI i = 1, . . . , m. Since y ∈ ω(x, T ), there exists r ∈ N such that T r x ∈ U and r̄ (m) + sα ∈ Nm . Then r̄ (m) + sα ∈ N(x, U)m . Therefore, N(x, U) is a J-set. Proposition 8.6. Let (X, T ) be an invertible dynamical system, x ∈ X and n ∈ N. Then x is J -recurrent in (X, T ) if and only if it is J recurrent in (X, T n ). Proof. Without loss of generality, assume that Orb(x, T ) = X. Since J is multiplication invariant, if x is J -recurrent in (X, T n ), then so is in (X, T ). Conversely, if x is J -recurrent in (X, T ), then (X, T ) satisfies the multiple IP-recurrence property, so does (X, T n ). Since the interior of Orb(x, T n ) is dense in Orb(x, T n ), it is easy to see that (Orb(x, T n ), T n ) also satisfies the multiple IP-recurrence property. Then x is J -recurrent in (X, T n ). Proof of Lemma 8.1. It follows from Theorem 6.4, Theorem 6.3, Lemma 7.3 and Proposition 8.6. Proof of Theorem 8.2. Since F is a C-set, there exists an idempotent p ∈ h(J ) such that F ∈ p. Let x = 1F ∈ {0, 1}Z and y = px ∈ [1]. Then y is J -recurrent, x is strongly proximal to y and N(x, [1]) = F . (m) (1) Let {sφ(α) = (sα , . . . , sα )} be an IP-system in Zm . Let U1 = [1]. Since N((x, y), U1 × U1 ) is a J-set, there exists r1 ∈ N and α1 ∈ Pf (N) (i) such that σ × σ r1 +sα1 (x, y) ∈ U1 × U1 for i = 1, 2, . . . , m. By continuity of σ, choose a neighborhood U2 of y such that U2 ⊂ U1 and m [ (i) σ r1 +sα1 U2 ⊂ U1 . i=1 Now suppose that we have choose neighborhood U1 , U2 , . . . Un , Un+1 of y, r1 , r2 , . . . , rn in N and α1 < α2 < · · · < αn in Pf (N) satisfying P the following conditions. For every β ⊂ {1, 2, . . . , n}, let rβ = j∈β rj , S φ(β) = j∈β αj and Uβ = Umin β , we have (1) σ (i) rβ +sφ(β) x ∈ Uβ for i = 1, . . . , m. (i) rβ +sφ(β) (2) σ Un+1 ⊂ Uβ for i = 1, . . . , m Since N((x, y), Un+1 × Un+1 ) is a J-set, there exists rn+1 ∈ N and (i) αn+1 > αn such that σ × σ rn+1 +sαn+1 (x, y) ∈ Un+1 × Un+1 for i = 1, 2, . . . , m. Choose a neighborhood Un+2 of y such that Un+2 ⊂ Un+1 and m [ (i) σ rn+1 +sαn+1 Un+2 ⊂ Un+1 . i=1 DYNAMICAL CHARACTERIZATION OF C-SETS AND ITS APPLICATION 29 Now we show that (1) and (2) are satisfied with β replaced by β ′ = β ∪ {n + 1} and n + 1 replaced by n + 2. This in fact follows from (i) σ rβ ′ +sφ(β ′ ) (i) (i) (i) x ∈ σ rβ +sφ(β) (σ rn+1 +sαn+1 x) ∈ σ rβ +sφ(β) Un+1 ⊂ Uβ and (i) σ rβ ′ +sφ(β ′ ) Un+2 ⊂ σ (i) rβ +sφ(β) (i) (σ rn+1 +sαn+1 Un+2 ) ⊂ σ r +s (i) rβ +sφ(β) Un+1 ⊂ Uβ . (i) Then by induction, we have that σ β φ(β) x ∈ [1] for every β ∈ Pf (N) (m) and i = 1, 2, . . . , m. Thus, we obtain that for every α ∈ Pf (N), r̄α + sφ(α) ∈ F m . Remark 8.7. One can use the algebraic properties of βN to prove Theorem 8.2 ([3, 20]). It is interesting that whether we can prove Lemma 8.1 by algebraic properties of βN. Acknowledgement. The author would like to thank Prof. Xiangdong Ye for helpful suggestions which significantly improved the presentation. This work was partly supported by the National Natural Science Foundation of China (Nos 11171320, 11001071, 11071231). References [1] E. Akin, Recurrence in Topological Dynamical Systems: Families and Ellis Actions, The University Series in Mathematics. Plenum Press, New York, 1997. [2] J. Banks, Regular periodic decompositions for topologically transitive maps, Ergod. Th. & Dynam. Sys. (1997), 17, 505–529. [3] M. Beiglbock, V. Bergelson, T. Downarowicz, and A. Fish, Solvability of Rado Systems in D-sets, Topology and its Applications 156 (2009), 2565–2571. [4] V. Bergelson, T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure preserving systems, Colloq. Math. 110 (2008), 117–150. [5] V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey Theory, Trans. Amer. Math. Soc. 320 (1990), 293–320. [6] V. Bergelson, N. Hindman and B. Kra, Iterated spectra of numberselementary, dynamical, and algebraic approaches, Trans. Amer. Math. Soc. 348 (1996), no. 3, 893–912. [7] A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130, (2002), no. 12, 3571–3578. [8] S. Burns and N. Hindman, Quasi-central sets and their dynamical characterization, Topology Proceedings 31 (2007), 445–455. [9] D. De, N. Hindman and D. Strauss, A new and stronger central sets theorem, Fund. Math. 199 (2) (2008), 155–175. [10] R. Ellis, Lectures on topological dynamics, Benjamin, New York, 1969. [11] R. Ellis and H. Keynes, Bohr compactifications and a result of Følner, Israel J. Math, 12 (1972), 314–330. 30 J. LI [12] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. [13] H. Furstenberg, Y. Katznelson, An ergodic Szemeredi theorem for IP-systems and combinatorial theory, J. Anal. Math. 45 (1985) 117–168. [14] H. Furstenberg, B. Weiss, Topological dynamics combinatorial number theory, J. Anal. Math. 34 (1978) 61–85. [15] S. Glasner, Divisible properties and the Stone-Čech compactification, Can. J. Math., 34 (1980), no. 4, 993–1007. [16] A. Hales and R. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229. [17] N. Hindman, Small sets satisfying the central sets theorem, in: Combinatorial number theory, B. Landman etc (ed.), Walter de Gruyter, Berlin, 2009, 57–63. [18] N. Hindman, A. Maleki, and D. Strauss, Central sets and their combinatorial characterization, J. Comb. Theory (Series A) 74 (1996), 188–208. [19] N. Hindman and D. Strauss, Algebra in the Stone-Čech compactification: Theory and applications, de Gruyter, Berlin, 1998. [20] N. Hindman, D. Strauss, Sets satisfying the central sets theorem, Semigroup Forum 79 (2009), no. 3, 480–506. [21] N. Hindman and D. Strauss, Cartesian products of sets satisfying the Central Sets Theorem, Topology Proceedings 35 (2010), 203–223. [22] W. Huang, H. Li and X. Ye, Family-independence for topological and measurable dynamics, Trans. Amer. Math. Soc., in press. http://arxiv.org/abs/0908.0574 [23] W. Huang, K. K. Park amd X. Ye, Dynamical systems disjoint from all minimal systems with zero entropy, Bull. Soc. Math. France, 135 (2007), 259–282. [24] W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357, (2005), no. 2, 669–694. [25] J. Li, Transitive points via Furstenberg family, Topology and its Applications, 158 (2011), 2221–2231. [26] R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1) (1933), 424–470. [27] S. Shao, Dynamical systems and Furstenberg families, University of Science and Technology of China, doctoral dissertation, 2004. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China E-mail address: [email protected]